hs_vectorization_isometry_intertwines_rank_one_tangent

The Hilbert-Schmidt vectorization $\mathrm{vec}(X)_{(a,b)} = X_{ab}$ flattening an $n_1\times n_2$ matrix to a vector indexed by coordinate pairs is a Frobenius-to-Euclidean isometry, and it intertwines each rank-one tangent operator with bot6 vecMulVec rank-one matrix. Concretely the theorem asserts three facts. (A) The Frobenius inner product equals the dot product of vectorizations: $\langle X,Y\rangle_F = \mathrm{vec}(X)\cdot\mathrm{vec}(Y)$. (B) The squared Frobenius norm equals the self dot product of the vectorization. (C) For $y_{ab} = P_T(e_a e_b^*) = \mathrm{tangentProjection}\,S\,(\mathrm{coordinateMatrix}\,a\,b)$, applying the matrix $\mathrm{vecMulVec}(\mathrm{vec}\,y_{ab})(\mathrm{vec}\,y_{ab})$ to $\mathrm{vec}(H)$ equals $\langle y_{ab},H\rangle_F\cdot\mathrm{vec}(y_{ab})$, i.e. the rank-one Hilbert-Schmidt operator $H\mapsto\langle y_{ab},H\rangle_F\,y_{ab}$ (the Rudelson summand $y_{ab}\otimes y_{ab}$) vectorizes to vecMulVec. This is the linchpin that brings the rank-one tensor-operator primitives and the matrix Khintchine / symmetrization machinery to bear on the Candes-Recht tangent sampling deviation.